Cardinal inequalities for $S(n)$-spaces

Hajnal and Juh\'asz proved that if $X$ is a $T_1$-space, then $|X|\le 2^{s(X)\psi(X)}$, and if $X$ is a Hausdorff space, then $|X|\le 2^{c(X)\chi(X)}$ and $|X|\le 2^{2^{s(X)}}$. Schr\"oder sharpened the first two estimations by showing that if $X$ is a Hausdorff space, then $|X|\le 2^{Us(X)\psi_c(X)}$, and if $X$ is a Urysohn space, then $|X|\le 2^{Uc(X)\chi(X)}$. In this paper, for any positive integer $n$ and some topological spaces $X$, we define the cardinal functions $\chi_n(X)$, $\psi_n(X)$, $s_n(X)$, and $c_n(X)$, called respectively $S(n)$-character, $S(n)$-pseudocharacter, $S(n)$-spread, and $S(n)$-cellularity, and using these new cardinal functions we show that the above-mentioned inequalities could be extended to the class of $S(n)$-spaces. We recall that the $S(1)$-spaces are exactly the Hausdorff spaces and the $S(2)$-spaces are exactly the Urysohn spaces.